3D divergence theorem examples

Background

• 3D divergence theorem
  • Flux in three dimensions
    • Divergence
    • Triple integrals

The divergence theorem (quick recap)

• $\blueE{\textbf{F}}(x, y, z)$start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, left parenthesis, x, comma, y, comma, z, right parenthesis is some three-dimensional vector field.
• $\redE{V}$start color #bc2612, V, end color #bc2612 is a three-dimensional volume (think of a blob in space).
• $\redE{S}$start color #bc2612, S, end color #bc2612 is the surface of $\redE{V}$start color #bc2612, V, end color #bc2612.
• $\greenE{\hat{\textbf{n}}}$start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f is a function which gives unit normal vectors on the surface of $\redE{S}$start color #bc2612, S, end color #bc2612.

Strategizing

Example 1: Surface integral through a cube.

Choose 1 answer:

Choose 1 answer:

• (Choice A)  

  $\displaystyle \int_0^1 \int_0^1 \int_0^1 \nabla \times (2y\hat{\textbf{i}} + 3y^2 \hat{\textbf{j}} + 4z\hat{\textbf{k}}) \;dx\,dy\,dz$integral, start subscript, 0, end subscript, start superscript, 1, end superscript, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, del, times, left parenthesis, 2, y, start bold text, i, end bold text, with, hat, on top, plus, 3, y, squared, start bold text, j, end bold text, with, hat, on top, plus, 4, z, start bold text, k, end bold text, with, hat, on top, right parenthesis, d, x, d, y, d, z

  A

  $\displaystyle \int_0^1 \int_0^1 \int_0^1 \nabla \times (2y\hat{\textbf{i}} + 3y^2 \hat{\textbf{j}} + 4z\hat{\textbf{k}}) \;dx\,dy\,dz$integral, start subscript, 0, end subscript, start superscript, 1, end superscript, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, del, times, left parenthesis, 2, y, start bold text, i, end bold text, with, hat, on top, plus, 3, y, squared, start bold text, j, end bold text, with, hat, on top, plus, 4, z, start bold text, k, end bold text, with, hat, on top, right parenthesis, d, x, d, y, d, z
• (Choice B)  

  $\displaystyle \int_0^1 \int_0^1 \int_0^1 \nabla \cdot (2y\hat{\textbf{i}} + 3y^2 \hat{\textbf{j}} + 4z\hat{\textbf{k}}) \;dx\,dy\,dz$integral, start subscript, 0, end subscript, start superscript, 1, end superscript, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, del, dot, left parenthesis, 2, y, start bold text, i, end bold text, with, hat, on top, plus, 3, y, squared, start bold text, j, end bold text, with, hat, on top, plus, 4, z, start bold text, k, end bold text, with, hat, on top, right parenthesis, d, x, d, y, d, z

  B

  $\displaystyle \int_0^1 \int_0^1 \int_0^1 \nabla \cdot (2y\hat{\textbf{i}} + 3y^2 \hat{\textbf{j}} + 4z\hat{\textbf{k}}) \;dx\,dy\,dz$integral, start subscript, 0, end subscript, start superscript, 1, end superscript, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, del, dot, left parenthesis, 2, y, start bold text, i, end bold text, with, hat, on top, plus, 3, y, squared, start bold text, j, end bold text, with, hat, on top, plus, 4, z, start bold text, k, end bold text, with, hat, on top, right parenthesis, d, x, d, y, d, z
• (Choice C)  

  $\displaystyle 6 \int_0^1 \int_0^1 \nabla \times (2y\hat{\textbf{i}} + 3y^2 \hat{\textbf{j}} + 4z\hat{\textbf{k}}) \;dx\,dy$6, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, del, times, left parenthesis, 2, y, start bold text, i, end bold text, with, hat, on top, plus, 3, y, squared, start bold text, j, end bold text, with, hat, on top, plus, 4, z, start bold text, k, end bold text, with, hat, on top, right parenthesis, d, x, d, y

  C

  $\displaystyle 6 \int_0^1 \int_0^1 \nabla \times (2y\hat{\textbf{i}} + 3y^2 \hat{\textbf{j}} + 4z\hat{\textbf{k}}) \;dx\,dy$6, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, del, times, left parenthesis, 2, y, start bold text, i, end bold text, with, hat, on top, plus, 3, y, squared, start bold text, j, end bold text, with, hat, on top, plus, 4, z, start bold text, k, end bold text, with, hat, on top, right parenthesis, d, x, d, y
• (Choice D)  

  $\displaystyle 6 \int_0^1 \int_0^1 \nabla \cdot (2y\hat{\textbf{i}} + 3y^2 \hat{\textbf{j}} + 4z\hat{\textbf{k}}) \;dx\,dy$6, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, del, dot, left parenthesis, 2, y, start bold text, i, end bold text, with, hat, on top, plus, 3, y, squared, start bold text, j, end bold text, with, hat, on top, plus, 4, z, start bold text, k, end bold text, with, hat, on top, right parenthesis, d, x, d, y

  D

  $\displaystyle 6 \int_0^1 \int_0^1 \nabla \cdot (2y\hat{\textbf{i}} + 3y^2 \hat{\textbf{j}} + 4z\hat{\textbf{k}}) \;dx\,dy$6, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, integral, start subscript, 0, end subscript, start superscript, 1, end superscript, del, dot, left parenthesis, 2, y, start bold text, i, end bold text, with, hat, on top, plus, 3, y, squared, start bold text, j, end bold text, with, hat, on top, plus, 4, z, start bold text, k, end bold text, with, hat, on top, right parenthesis, d, x, d, y

Check

[Answer]

$\nabla \cdot (2y\hat{\textbf{i}} + 3y^2 \hat{\textbf{j}} + 4z\hat{\textbf{k}}) =$del, dot, left parenthesis, 2, y, start bold text, i, end bold text, with, hat, on top, plus, 3, y, squared, start bold text, j, end bold text, with, hat, on top, plus, 4, z, start bold text, k, end bold text, with, hat, on top, right parenthesis, equals

Check

[Answer]

Example 2: Surface integral through a cylinder

$\displaystyle \nabla \cdot \left[ \begin{array}{c} x^3 \\ y^3 \\ x^3 + y^3 \end{array} \right] =$

Check

[Answer]

$\displaystyle \iiint_{\redE{C}} \left( \nabla \cdot \left[ \begin{array}{c} x^3 \\ y^3 \\ x^3 + y^3 \end{array} \right] \right) \; \redE{dV} =$

Check

[Answer]

Example 3: Surface area from a volume integral

Choose 1 answer:

Choose 1 answer:

• (Choice A)  

  $\blueE{\textbf{F}} = \greenE{\hat{\textbf{n}}}$start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, equals, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f

  A

  $\blueE{\textbf{F}} = \greenE{\hat{\textbf{n}}}$start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, equals, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f
• (Choice B)  

  $\blueE{\textbf{F}} = \greenE{\hat{\textbf{n}}} \times ( x \hat{\textbf{i}} + y \hat{\textbf{j}} + z \hat{\textbf{k}})$start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, equals, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, times, left parenthesis, x, start bold text, i, end bold text, with, hat, on top, plus, y, start bold text, j, end bold text, with, hat, on top, plus, z, start bold text, k, end bold text, with, hat, on top, right parenthesis

  B

  $\blueE{\textbf{F}} = \greenE{\hat{\textbf{n}}} \times ( x \hat{\textbf{i}} + y \hat{\textbf{j}} + z \hat{\textbf{k}})$start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, equals, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, times, left parenthesis, x, start bold text, i, end bold text, with, hat, on top, plus, y, start bold text, j, end bold text, with, hat, on top, plus, z, start bold text, k, end bold text, with, hat, on top, right parenthesis

Check

[Answer]

Choose 1 answer:

Choose 1 answer:

• (Choice A)  

  $\displaystyle \iiint_\redE{B} \nabla \cdot \greenE{\hat{\textbf{n}}} \,\redE{dV}$\iiint, start subscript, start color #bc2612, B, end color #bc2612, end subscript, del, dot, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, start color #bc2612, d, V, end color #bc2612

  A

  $\displaystyle \iiint_\redE{B} \nabla \cdot \greenE{\hat{\textbf{n}}} \,\redE{dV}$\iiint, start subscript, start color #bc2612, B, end color #bc2612, end subscript, del, dot, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, start color #bc2612, d, V, end color #bc2612
• (Choice B)  

  $\displaystyle \iiint_\redE{B} \nabla \cdot (\greenE{\hat{\textbf{n}}} \cdot \greenE{\hat{\textbf{n}}}) \,\redE{dV}$\iiint, start subscript, start color #bc2612, B, end color #bc2612, end subscript, del, dot, left parenthesis, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, dot, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, right parenthesis, start color #bc2612, d, V, end color #bc2612

  B

  $\displaystyle \iiint_\redE{B} \nabla \cdot (\greenE{\hat{\textbf{n}}} \cdot \greenE{\hat{\textbf{n}}}) \,\redE{dV}$\iiint, start subscript, start color #bc2612, B, end color #bc2612, end subscript, del, dot, left parenthesis, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, dot, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, right parenthesis, start color #bc2612, d, V, end color #bc2612

Check

[Answer]

Choose 1 answer:

Choose 1 answer:

• (Choice A)  

  $\displaystyle \greenE{\hat{\textbf{n}}}(x, y, z) = (x + y + z) (\hat{\textbf{i}} + \hat{\textbf{j}} + \hat{\textbf{k}})$start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, left parenthesis, x, comma, y, comma, z, right parenthesis, equals, left parenthesis, x, plus, y, plus, z, right parenthesis, left parenthesis, start bold text, i, end bold text, with, hat, on top, plus, start bold text, j, end bold text, with, hat, on top, plus, start bold text, k, end bold text, with, hat, on top, right parenthesis

  A

  $\displaystyle \greenE{\hat{\textbf{n}}}(x, y, z) = (x + y + z) (\hat{\textbf{i}} + \hat{\textbf{j}} + \hat{\textbf{k}})$start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, left parenthesis, x, comma, y, comma, z, right parenthesis, equals, left parenthesis, x, plus, y, plus, z, right parenthesis, left parenthesis, start bold text, i, end bold text, with, hat, on top, plus, start bold text, j, end bold text, with, hat, on top, plus, start bold text, k, end bold text, with, hat, on top, right parenthesis
• (Choice B)  

  $\displaystyle \greenE{\hat{\textbf{n}}}(x, y, z) = \dfrac{1}{\sqrt{3}}x \hat{\textbf{i}} + \dfrac{1}{\sqrt{3}}y \hat{\textbf{j}} + \dfrac{1}{\sqrt{3}}z \hat{\textbf{k}}$start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, left parenthesis, x, comma, y, comma, z, right parenthesis, equals, start fraction, 1, divided by, square root of, 3, end square root, end fraction, x, start bold text, i, end bold text, with, hat, on top, plus, start fraction, 1, divided by, square root of, 3, end square root, end fraction, y, start bold text, j, end bold text, with, hat, on top, plus, start fraction, 1, divided by, square root of, 3, end square root, end fraction, z, start bold text, k, end bold text, with, hat, on top

  B

  $\displaystyle \greenE{\hat{\textbf{n}}}(x, y, z) = \dfrac{1}{\sqrt{3}}x \hat{\textbf{i}} + \dfrac{1}{\sqrt{3}}y \hat{\textbf{j}} + \dfrac{1}{\sqrt{3}}z \hat{\textbf{k}}$start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, left parenthesis, x, comma, y, comma, z, right parenthesis, equals, start fraction, 1, divided by, square root of, 3, end square root, end fraction, x, start bold text, i, end bold text, with, hat, on top, plus, start fraction, 1, divided by, square root of, 3, end square root, end fraction, y, start bold text, j, end bold text, with, hat, on top, plus, start fraction, 1, divided by, square root of, 3, end square root, end fraction, z, start bold text, k, end bold text, with, hat, on top
• (Choice C)  

  $\displaystyle \greenE{\hat{\textbf{n}}}(x, y, z) = x \hat{\textbf{i}} + y \hat{\textbf{j}} + z \hat{\textbf{k}}$start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, left parenthesis, x, comma, y, comma, z, right parenthesis, equals, x, start bold text, i, end bold text, with, hat, on top, plus, y, start bold text, j, end bold text, with, hat, on top, plus, z, start bold text, k, end bold text, with, hat, on top

  C

  $\displaystyle \greenE{\hat{\textbf{n}}}(x, y, z) = x \hat{\textbf{i}} + y \hat{\textbf{j}} + z \hat{\textbf{k}}$start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, left parenthesis, x, comma, y, comma, z, right parenthesis, equals, x, start bold text, i, end bold text, with, hat, on top, plus, y, start bold text, j, end bold text, with, hat, on top, plus, z, start bold text, k, end bold text, with, hat, on top

Check

[Answer]

$\nabla \cdot \greenE{\hat{\textbf{n}}} =$del, dot, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, equals

Check

[Answer]

$\displaystyle \iiint_\redE{B} \nabla \cdot \greenE{\hat{\textbf{n}}} \,\redE{dV} =$\iiint, start subscript, start color #bc2612, B, end color #bc2612, end subscript, del, dot, start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, start color #bc2612, d, V, end color #bc2612, equals

Check

[Answer]

Summary

• The divergence theorem is useful whenever the interior volume of a region is easier to describe than its surface.
• It also helps if the divergence of the relevant vector field turns it into a simpler function.